Download Theoretical investigation of magnetic-field

PHYSICAL REVIEW A 88, 013416 (2013)
Theoretical investigation of magnetic-field-induced 2 p5 3s 3P0,2 – 2 p6 1S0 transitions
in Ne-like ions without nuclear spin
Jiguang Li,1 Jon Grumer,1 Wenxian Li,2,3 Martin Andersson,2,3 Tomas Brage,1
Roger Hutton,2,3 Per Jönsson,4 Yang Yang,2,3,* and Yaming Zou2,3
1
Department of Physics, Lund University, S-221 00 Lund, Sweden
The Key Laboratory of Applied Ion Beam Physics, Ministry of Education, Shanghai, China
3
Shanghai EBIT Laboratory, Institute of Modern Physics, Fudan University, Shanghai, China
4
Group for Materials Science and Applied Mathematics, Malmö University, S-205 06 Malmö, Sweden
(Received 7 March 2013; published 22 July 2013)
2
We report theoretical results for magnetic-field-induced 2p5 3s 3P0,2 − 2p 6 1S0 E1 transitions in Ne-like ions
with zero nuclear spin (I = 0) between Mg III and Zn XXI as well as in Ne I. We demonstrate that it is important to
include both “perturber” states 2p 5 3s 1P1 and 2p 5 3s 3P1 in order to produce reliable transition rates. Furthermore,
we investigate the trends of the rates along the isoelectronic sequence of the 2p5 3s 3P0,2 − 2p 6 1S0 transitions and
their competition with the 2p 5 3s 3P0 − 2p 5 3s 3P1 M1 and the 2p 5 3s 3P2 − 2p 6 1S0 M2 decays. For the 2p 5 3s 3P0
state the magnetic-field-induced transition becomes the dominant decay channel for the light elements even in a
relatively weak magnetic field, and it will therefore prove useful in diagnostics of the strength of magnetic fields
in different plasmas. The influence of an external magnetic field on the lifetime of the 2p5 3s 3P2 state is much
smaller but still observable for the ions near the neutral end of the sequence. As a special case, the magnetic
field effect on the lifetimes of 2p 5 3s 3P0,2 states of neutral 20 Ne is discussed. It is found that the lifetimes are
drastically reduced by a magnetic field, which may be an underlying reason for the discrepancies in the lifetime
of the 2p 5 3s 3P2 state between experiment [14.73(14) s] and theory (17.63 s).
DOI: 10.1103/PhysRevA.88.013416
PACS number(s): 32.60.+i, 31.15.ag
I. INTRODUCTION
The effects of magnetic fields are important in many
astrophysical or laboratory plasmas and their strengths are
crucial plasma parameters [1–3]. It is well known that the
interaction between the magnetic field and an atom (or ion)
causes spectral lines to split into groups of lines (Zeeman
splitting), which can be used to determine the magnetic field
strength in a plasma [2,3]. On the other hand, the magnetic
interaction also breaks the symmetry of an atomic system
allowing atomic states with the same magnetic quantum
number and parity to mix and bring about “unexpected” lines
to appear in spectra [4,5] and lifetimes of long lived state
to be shortened [6,7]. We will refer to these as magneticfield-induced transitions (MITs). In 2003, Beiersdorfer et al.
identified a magnetic-field-induced transition in Ne-like Ar
using the EBIT-II electron beam ion trap in the Lawrence
Livermore National Laboratory [8]. They illustrated that the
MIT can also be used for diagnostics of magnetic field strength
for high-temperature plasmas.
Considering the significance of the determination of the
magnetic field strengths in plasmas and the promising diagnostic method using MIT lines, we initiated a project to
systematically calculate the rates of magnetic-field-induced
2p5 3s 3P0,2 − 2p6 1S0 E1 transitions in Ne-like ions with
zero nuclear spin (I = 0) between Mg III and Zn XXI. The
transitions in neutral Ne, as a special case, are investigated
as well for the unresolved discrepancies in the lifetime
of the 2p5 3s 3P2 state [9]. For ions with nuclear spin the
hyperfine interaction also induces transitions from 2p5 3s 3P0,2
*
[email protected]
1050-2947/2013/88(1)/013416(8)
to 2p6 1S0 [9,10]. However, since the aim of this work is to
investigate the MITs, we do not consider these transitions
in the present paper. The calculations are performed with
GRASP2K [11] and HFSZEEMAN [12] packages based on
the multiconfiguration Dirac-Hartree-Fock (MCDHF) method
[13]. We emphasize the importance of different perturbers that
give rise to the MIT. In addition, the systematic behavior of
lines that dominate the decays of the 2p5 3s 3P0 and 2p5 3s 3P2
levels, namely, magnetic-field-induced 2p5 3s 3P0,2 − 2p6 1S0
E1, 2p5 3s 3P0 − 2p5 3s 3P1 M1, and 2p5 3s 3P2 − 2p6 1S0 M2
transitions (cf. Fig. 1) are investigated along the Ne I
isoelectronic sequence.
II. THEORETICAL METHODS AND
COMPUTATIONAL MODELS
A. General theory
In the presence of an external magnetic field B, the
Hamiltonian of an atom without nuclear spin is
H = Hf s + Hm ,
(1)
where Hf s in our approach is the relativistic fine-structure
Hamiltonian including the Breit interaction and parts of
quantum electrodynamical (QED) effects, and Hm is the
interaction Hamiltonian with the external magnetic field. If
the magnetic field is homogeneous through the atomic system,
the interaction Hamiltonian is expressed by [14]
Hm = (N(1) + N(1) ) · B,
(2)
where the last term is the so-called Schwinger QED correction. For an N -electron atom the tensor operators are
013416-1
©2013 American Physical Society
JIGUANG LI et al.
PHYSICAL REVIEW A 88, 013416 (2013)
Substituting Eq. (5) into Eq. (7), we have
2.02613 × 1018 dJ d J (−1)J −M
A=
λ3
q
J J
2
J
1 J
(1)
J
||P
×
||
J
(8)
.
−M q M FIG. 1. The level structure and transitions for the five lowest states
of a Ne-like system.
where λ is wavelength in angstroms, and the square of the
reduced matrix element of the electric dipole transition P(1)
operator is basically the line strength of the corresponding
transition (in a.u.) [15]. Since the magnetic interaction mixes
atomic states with different J by 1, the selection rule on the
total angular momentum for E1 transitions can be extended
to J = J − J = 0,±1,±2, and ±3. As an example, onephoton forbidden E1 J = 0 ↔ J = 0 transitions are induced
by an external magnetic field.
In the framework of the MCDHF method [13], atomic state
wave functions |J M can be expressed by a linear combination of configuration state functions (CSFs) with the same
parity P , total angular momentum J , and its z component M,
that is,
ci |γi J M,
(9)
|J M =
given by
i
N(1)
√
N
N
2
(1)
rk [α k C (1) (k)](1) ,
=
n (k) =
−i
2α
k=1
k=1
N(1) =
N
n(1) (k) =
k=1
N
gs − 2
βk k ,
2
k=1
(3)
(4)
where rk is the radial coordinate of the kth electron, C (1) (k)
is a spherical tensor operator of rank 1, k is the relativistic
spin matrix, and gs = 2.00232, which is the g factor of the
electron spin corrected for QED effects. In addition, α is the
fine-structure constant and α and β are Dirac matrices.
We choose the direction of the magnetic field as the
z direction, making M the only good quantum number. The
M-dependent atomic state wave function |M can be written
as an expansion
|M =
dJ |J M.
(5)
J
where the |J M are atomic state wave functions (ASFs) that
are eigenstates of the Hamiltonian Hf s . The coefficients dJ
are obtained either through perturbation theory or by solving
the eigenvalue equation using the HFSZEEMAN package [12].
In terms of perturbation theory, dJ is given, to first order, by
dJ =
J M|Hm |0 J0 M0 ,
E(0 J0 ) − E(J )
(6)
where |0 J0 M0 represents the reference atomic state.
For an atomic system in an external magnetic field, electric
dipole (E1) transition probabilities from a state |M to another
state |M are expressed in s−1 by
A=
2
2.02613 × 1018 M|Pq(1) |M .
3
λ
q
(7)
where ci stands for the mixing coefficient, and the label is often the same as the label γi of the dominating CSF.
The latter denotes other appropriate quantum numbers of
the CSFs, such as orbital occupancy and coupling tree.
CSFs are built up from products of one-electron relativistic
orbitals. Applying the variational principle, one is able to
obtain one-electron relativistic orbitals and mixing coefficients
from a self-consistent field (SCF) procedure. The Breit
interaction
(α i · r ij )(α j · r ij )
1
Bij = −
αi · αj +
(10)
2rij
rij2
and leading QED effects are taken into account in subsequent
relativistic configuration interaction (RCI) computations.
B. MITs in Ne-like ions
For Ne-like ionic systems in an external magnetic field, the
reference states 2p5 3s 3P0,2 are approximately expressed as
|“2p5 3s 3P0,2 ”M = d0 |2p5 3s 3P0,2 M
+
dS;J =1 |2p5 3s SP1 M.
(11)
S(=1,3)
The quotation marks on the left-hand side emphasize the
fact that the notation is just a label indicating the dominant component of the eigenvector. It should be pointed
out that the mixing coefficients d are different between
the |“2p5 3s 3P0 ”M and |“2p5 3s 3P2 ”M states. Remaining
interactions between 2p5 3s 3P0,2 and other atomic states are
neglected due to large energy separations and comparatively
weak magnetic interaction couplings. The ground state is
very well isolated from other states and it is, therefore, a
good approximation to assume that its wave function can be
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THEORETICAL INVESTIGATION OF MAGNETIC-FIELD- . . .
PHYSICAL REVIEW A 88, 013416 (2013)
TABLE I. Off-diagonal reduced matrix elements W (in a.u.) of the magnetic interaction together with reduced mixing coefficients dSR
(in T−1 ) for Ne-like Mg and Zn ions. DHF: single configuration Dirac-Hartree-Fock; RCI: relativistic configuration interaction; BI: Breit
interaction; QED: quantum electrodynamics effects. Numbers in square brackets are the powers of 10.
(3P0 ,3P1 )
Model
(3P0 ,1P1 )
d1R
W
(3P2 ,3P1 )
W
d3R
W
d1R
W
d3R
−0.3440
−0.3440
−0.3453
−0.3453
2.4997[−4]
2.5532[−4]
2.6411[−4]
2.6951[−4]
−0.08523
−0.08522
−0.07971
−0.07989
1.3281[−5]
1.3379[−5]
1.2694[−5]
1.2716[−5]
−0.2273
−0.2267
−0.2269
−0.2268
1.0671[−5]
1.0933[−5]
1.0869[−5]
1.0865[−5]
−0.2717
−0.2721
−0.2720
−0.2721
1.1991[−6]
1.2055[−6]
1.2136[−6]
1.2155[−6]
DHF
RCI
BI
QED
−0.3972
−0.3972
−0.3987
−0.3986
−3.5252[−4]
−3.3507[−4]
−3.6263[−4]
−3.6161[−4]
−0.09869
−0.09869
−0.09234
−0.09254
Mg2+
2.5210[−5]
2.5721[−5]
2.3667[−5]
2.3734[−5]
DHF
RCI
BI
QED
−0.2611
−0.2604
−0.2605
−0.2605
−1.3550[−6]
−1.3499[−6]
−1.3601[−6]
−1.3569[−6]
−0.3157
−0.3162
−0.3161
−0.3161
Zn20+
2.5105[−5]
2.6076[−5]
2.6990[−5]
2.6995[−5]
written as
|“2p6 1S0 ”M = |2p6 1S0 M.
(12)
5
1
The mixing with the “perturber” states of 2p 3s P1 and
2p5 3s 3P1 in Eq. (11) opens up the one-photon 2p5 3s 3P0,2 −
2p6 1S0 E1 transitions. Inserting the angular quantum numbers
for the states into Eq. (8) and evaluating the 3-j symbol gives
the magnetically induced transition rate AMIT as
AMIT =
2.02613 × 1018
3λ3
2
61
(1)
5
S
×
dS 2p S0 ||P ||2p 3s P1 .
S(=1,3)
(13)
For weak to moderate magnetic fields where Eq. (6) from
the first-order perturbation theory holds, dS is proportional to
the magnetic field strength B, and we can define a reduced
coefficient dSR and reduced transition rate ARMIT , which are
independent of B through
dS = dSR B, AMIT = ARMIT B 2 .
(14)
C. Computational models
The calculations are carried out by using the same computational strategy as described in Refs. [16,17]. The active set
method is used to construct the configuration space. Configuration expansions are generated by single (S) and double (D)
replacements of orbitals in the reference configurations with
ones in an active set. In the present work, a single reference
configuration model is adopted as a starting description for
the ground and excited states, and the 1s core shell is kept
closed. The configuration spaces are therefore obtained by SD
R=
(3P2 ,1P1 )
excitations from the remaining shells of the single reference
configurations to the active set. The active set is augmented
layer by layer until n = 7. We impose the restriction on the
expansion at the last step (n = 7) for the excited states that we
allow at most one excitation from 2s or 2p. Considering the
stability problems in the SCF procedure we optimize only the
orbitals in the last added correlation layer at the time (together
with mixing coefficients). The RCI computations following the
SCF calculations take into account the residual correlations as
well as the Breit interaction and the QED corrections, since
the configuration spaces are further expanded by including the
CSFs obtained from all SD excitations to the n = 7 orbital set
and with triple (T) excitations up to the n = 4 orbital set. With
this model, high quality atomic state wave functions based
on expansions of several hundred thousand CSFs have been
produced, which give excellent transition energies with only
0.011% errors compared with highly accurate measurements
available. To further check the effect of electron correlations,
the Breit interaction, and the QED effects on the magnetic
interaction, we present in Table I off-diagonal reduced matrix
elements W = J ||N(1) + N(1) || J for 3P0 and 3P2 states
in the cases of Mg III and Zn XXI as well as reduced associated
mixing coefficients dSR . It is worth noting that the Breit
interaction affects the off-diagonal reduced matrix elements at
the low-Z end to some extent. As the atomic number increases,
the Breit interaction becomes less relatively important and
thus the effect decreases. On the other hand, the mixing
coefficients are influenced by both electron correlation and
Breit interaction effects, since fine structure splittings in the
2p5 3s configuration are sensitive to these.
An earlier estimate of the MIT rate for Ne-like Ar has only
included one perturbating state [8], but we find that this model
is not sufficient and illustrate in Fig. 2 the relative importance
of the two included perturbers in Eq. (13) through the ratio
2d3 d1 2p6 1S0 ||P(1) ||2p5 3s 3P1 2p6 1S0 ||P(1) ||2p5 3s 1P1 (d3 2p6 1S0 ||P(1) ||2p5 3s 3P1 )2 + (d1 2p6 1S0 ||P(1) ||2p5 3s 1P1 )2
013416-3
(15)
JIGUANG LI et al.
PHYSICAL REVIEW A 88, 013416 (2013)
14 < Z < 20. It is clear that the two contributions are of comparable size for all ions under investigation and thus both have
to be included for accurate results. With respect to the sign of
R, we find that the contribution from 2p5 3s 3P1 and 2p5 3s 1P1
perturbers are constructive for the magnetic-field-induced
2p5 3s 3P0 − 2p6 1S0 E1 transition and are destructive for the
magnetic-field-induced 2p5 3s 3P2 − 2p6 1S0 E1 transition.
D. Breit-Pauli estimates of MIT rates
In this subsection, we confirm our Dirac-Hartree-Fock and
Breit results by a Hartree-Fock (HF) calculation including relativistic corrections through the Breit-Pauli (BP) approximation
for the magnetic-field-induced 2p5 3s 3P0 − 2p6 1S0 transition
rate of Mg2+ . Including the four main CSFs, the resulting
atomic state wave functions labeled by the largest component
can be obtained with the ATSP2K package [18]:
FIG. 2. (Color online) The R values for the magnetic-field
2p 5 3s 3P0,2 − 2p 6 1S0 transitions.
|“2p5 3s 3P0 ” = |3P0 ,
|“2p5 3s 3P1 ” = +0.9738|3P1 − 0.2275|1P1 ,
as a function of the atomic number Z. The value of R varies
between 1 and −1, depending on the relative phase between
two terms involved in the transition amplitudes of MITs. If
R > 0, these two perturbers make a constructive contribution
to the MIT rate. Otherwise, there will be a cancellation between
perturbers in the rate. When |R| = 1, two perturbers are
of equivalent importance and thus must both be taken into
account. As can be seen from Fig. 2, the values of R vary
nonmonotonously along the isoelectronic sequence for these
two MITs, and are close to 1, especially for the ions with
|“2p5 3s 3P2 ” = |3P2 ,
|“2p5 3s 1P1 ” = +0.9738|1P1 + 0.2275|3P1 .
Using this basis, atomic parameters involved in the MIT, such
as the transition energies and line strengths, are calculated and
listed in Table II. The magnetic interaction Hamiltonian Hm
in the the nonrelativistic approximation, assuming a magnetic
field in the z direction, is Hm = μB B(Lz + gs Sz ) [19]. The offdiagonal matrix elements between the CSFs can be expressed
as [20]
γ LSJ MJ |Lz + gs Sz |γ L S J MJ = δγ γ δLL δSS δJ,J (=J −1) gJ,J (=J −1) (LS)(J 2 − M 2 )1/2 ,
where
gJ,J −1 (LS) = −(gs − 1) ×
(17)
(J + L + S + 1)(J + L − S)(J + S − L)(L + S − J + 1)
,
4J 2 (2J − 1)(2J + 1)
with gs = 2.00232. The L and S in the equations above
stand for the total orbital and spin angular momentums and Lz and Sz are their components along the
z direction. Hence we can write the matrix elements
in the relation for the reduced mixing coefficients dSR
(16)
(18)
[see Eqs. (6) and (11)] as
“2p5 3s SP1 ”|Lz + gs Sz |“2p5 3s 3P0 ”
E(“2p5 3s 3P0 ”) − E(“2p5 3s SP1 ”)
μB
gJ,J −1 (LS),
= c(3P1 )
E
dSR = μB
(19)
TABLE II. The reduced magnetic-field-induced 2p5 3s 3P0 − 2p 6 1S0 E1 transition rate ARMIT (in s−1 T−2 ) in conjunction with the reduced
mixing coefficients dSR (in T−1 ), the reduced matrix element 2p6 1S0 ||P(1) ||2p 5 3s SP1 in the length gauge of the electric dipole transition P(1)
operator (in a.u.) and the wavelength λ (in angstroms) of the MIT for Ne-like Mg. The values marked with HF-BP and DHF-BI are obtained
from a Hartree-Fock calculation including relativistic corrections through the Breit-Pauli approximation and a Dirac-Hartree-Fock calculation
with the Breit interaction, respectively. Numbers in square brackets are the powers of 10.
Method
d3R
2p 6 1S0 ||P(1) ||2p 5 3s 3P1 d1R
2p 6 1S0 ||P(1) ||2p 5 3s 1P1 λ
ARMIT
HF-BP
DHF-BI
−3.801[−4]
−3.626[−4]
−0.0865
−0.0844
2.351[−5]
2.367[−5]
0.370
0.371
239
239
82
78
013416-4
THEORETICAL INVESTIGATION OF MAGNETIC-FIELD- . . .
where c(3P1 ) is the content of 3P1 in the “2p5 3s 3P1 ”
and “2p5 3s 1P1 ” states, and E = E(“2p5 3s 3P0 ”) −
E(“2p5 3s SP1 ”) and L = S = J = 1 for the case under investigation. The resulting mixing coefficients can be found in
Table II.
As can be seen from Table II, all atomic parameters
obtained with Hartree-Fock calculations including relativistic
corrections through the Breit-Pauli approximation are in
good agreement with Dirac-Hartree-Fock calculations with
the Breit interaction. Substituting these atomic data into Eq.
(13), we obtain the reduced rate, ARMIT = 82 s−1 T−2 , for the
2p5 3s 3P0 − 2p6 1S0 MIT in the case of Ne-like Mg, which is
in good consistency with the full relativistic calculation.
E. The estimates of uncertainties
There are two main error sources in the present calculations.
One is related to the quality of atomic state wave functions,
and the other is the perturbative treatment of the interaction
between an external magnetic field and an atomic system.
For the former the uncertainties result from the neglected
electron correlations and other effects, such as the frequencydependent Breit interaction (FDBI). We estimate that the
residual correlations and the FDBI effects contribute to the
off-diagonal matrix elements and the mixing coefficients
by about 3%. The perturbative treatment of the magnetic
interaction brings about fractional errors which are much less
than 1% for the MIT rates. Therefore, the total uncertainties
in the MIT rates are less than 5%.
III. RESULTS AND DISCUSSIONS
A. The magnetic-field-induced 2 p5 3s 3P0 − 2 p6 1S0 E1
transition
The reduced MIT rates [see Eq. (14)] are reported in
Table III, with the inclusion of only the 1P1 perturber, as well
as of both 1P1 and 3P1 . It is clear that the contribution from the
2p5 3s 3P1 is indeed significant to the MIT probabilities for a
major part of the sequence. Using the reduced total MIT rates,
one can readily obtain the MIT probabilities by Eq. (14) for any
magnetic field, but we give the values for some examples in
Table IV. With respect to plasma diagnostics, only the rate for
the most abundant isotope of Ne-like ions without nuclear spin
are given. In Ref. [8] Beiersdorfer et al. calculated the MIT
rate for Ne-like Ar and obtained a value of 2440 s−1 in the case
of B = 3 T. Using the same magnetic field strength, we have
a MIT rate of 3004 s−1 . The difference between their result
and ours are surprisingly small considering the fact that they
neglected the 2p5 3s 3P1 perturber. It can be seen from Table III
that the inclusion of this perturber increases the MIT rate by a
factor of 3 for Ne-like Ar. We believe that other uncertainties in
their calculations cancel the effect of neglecting one perturber.
In the last column of Table IV we also present the rates of
the 2p5 3s 3P0 − 2p5 3s 3P1 M1 transition. It is clear that the
MIT is a dominant decay channel of the 2p5 3s 3P0 state for
low-Z ions even in relatively weak external magnetic fields.
AMIT
The branching ratios of the MIT channel ( AMIT
) along the
+AM1
Ne-like isoelectronic sequence are given in Fig. 3 for some
magnetic fields. This property is also related to the relative
intensity between the MIT line and the 2p5 3s 1P1 − 2p6 1S0 E1
line (3F) [8]. Regarding plasma diagnostics it is interesting to
PHYSICAL REVIEW A 88, 013416 (2013)
TABLE IV. Magnetic-field-induced 2p5 3s 3P0 − 2p 6 1S0 E1 transition probabilities AMIT (in s−1 ) for abundant isotopes without
nuclear spin of Ne-like ions between Mg III and Zn XXI. AMIT are
given for magnetic fields B = 0.5,1.5, and 2.5 T through Eq. (14).
Magnetic dipole 3P0 − 3P1 transitions rates AM1 (in s−1 ) are presented
for convenience.
AMIT
Ions
24
Mg2+
Si4+
32 6+
S
40
Ar8+
40
Ca10+
48 12+
Ti
52
Cr14+
56
Fe16+
58
Ni18+
64
Zn20+
28
B = 0.5
B = 1.5
B = 2.5
AM1
26
42
61
83
109
140
174
212
257
307
235
378
549
751
985
1256
1563
1912
2309
2764
653
1049
1526
2086
2736
3488
4341
5312
6415
7677
0.0558
0.764
7.68
56.1
308
1349
4955
15880
45672
120249
note for which magnetic field the MIT rate equals the M1
rate. The corresponding critical magnetic field strength can be
calculated using
AM1
critical
B
=
.
(20)
ARMIT
This field is depicted in Fig. 4 as a function of Z. As can be
seen from this figure, the magnetic field covers a wide range
from a few thousand gauss to several teslas between Mg III and
Zn XXI.
B. The magnetic-field-induced 2 p5 3s 3P2 − 2 p6 1S0 E1 transition
Resembling the magnetic-field-induced 2p5 3s 3P0 −
2p6 1S0 E1 transition, one-photon E1 decay channels are
opened by the external magnetic field for the 2p5 3s 3P2
state with M = +1, 0, and − 1. The magnetic-field-induced
2p5 3s 3P2 − 2p6 1S0 transition rates are dependent on the
FIG. 3. (Color online) The branching ratios of the MIT for the
2p 5 3s 3P0 state under circumstances of B = 0.5, 1, 2, and 3 T,
respectively.
013416-5
JIGUANG LI et al.
PHYSICAL REVIEW A 88, 013416 (2013)
TABLE V. Reduced magnetic-field-induced 2p5 3s 3P2 −
2p S0 E1 transition probabilities ARMIT (M) (in s−1 T−2 ) for Ne-like
ions with 12 Z 30. ARMIT (M) are obtained by the inclusion of
both the 2p 5 3s 1P1 and the 2p 5 3s 3P1 perturbers. M represents the
magnetic quantum numbers of sublevels in the 2p5 3s 3P2 state. MIT
rates for any magnetic field can be obtained with ARMIT (M) through
AMIT (M) = ARMIT (M)B 2
61
FIG. 4. The strength of critical magnetic fields B critical in T [see
Eq. (20)].
magnetic quantum number M of sublevels belonging to the
2p5 3s 3P2 state through the mixing coefficients dS in Eq. (6).
For the 2p5 3s 3P2 state, dS is given by
SP1 M|Hm |3P2 M
E(3P2 ) − E(SP1 )
4 − M 2 3P2 ||N(1) + N(1) ||SP1 .
=B
6
E(3P2 ) − E(SP1 )
dS (M) =
(21)
As a result, the rates of the 2p5 3s 3P2 − 2p6 1S0 MITs for
individual sublevels can be expressed as
AMIT (M)
2.02613 × 1018 B 2 (4 − M 2 )
=
3λ3
6
2
3P2 ||N(1) + N(1) ||SP1 1
(1) S
×
S0 ||P || P1 .
3P ) − E(SP )
E(
2
1
S(=3,1)
Ions
M=0
M = ±1
Ions
M=0
M = ±1
Mg2+
Al3+
Si4+
P5+
S6+
Cl7+
Ar8+
K9+
Ca10+
Sc11+
15
18
22
26
31
36
41
47
53
60
11
14
16
19
23
27
31
35
40
45
Ti12+
V13+
Cr14+
Mn15+
Fe16+
Co17+
Ni18+
Cu19+
Zn20+
67
74
81
89
98
106
114
124
134
50
55
61
67
73
80
86
93
100
The two equations above show that the definition of the
reduced MIT rate [see Eq. (14)] is still valid for this transition.
Furthermore, AMIT (M) is proportional to (4 − M 2 )B 2 . In
Table V we report the reduced MIT rates from each magnetic
sublevel in the 2p5 3s 3P2 state to the ground state, which can
be used to calculate the MIT rates in a certain magnetic field.
In the absence of an external magnetic field, the magnetic
quadrupole (M2) 2p5 3s 3P2 − 2p6 1S0 transition is mainly
a one-photon decay channel. Hence the lifetime for the
2p5 3s 3P2 state is given by
τ=
1
.
AM2
(23)
When the external magnetic field is introduced, the MIT should
be taken into account and thus the lifetimes are dependent on
the M of sublevels in 2p5 3s 3P2 state. For each sublevel, we
have
τ (M) =
(22)
1
.
AM2 + AMIT (M)
(24)
TABLE III. Reduced magnetic-field-induced 2p5 3s 3P0 − 2p 6 1S0 E1 transition probabilities ARMIT (in s−1 T−2 ) for Ne-like ions with
12 Z 30. ARMIT are obtained by the inclusion of only the 2p5 3s 1P1 perturber (labeled “only 1P1 ”) and of both the 2p 5 3s 1P1 and the
2p 5 3s 3P1 perturbers (labeled “Total”), respectively. For comparison, 2p5 3s 3P0 − 2p 5 3s 3P1 M1 transition probabilities (in s−1 ) are displayed.
The numbers in square brackets represent the powers of 10. MIT rates for any magnetic field can be obtained with ARMIT through AMIT = ARMIT B 2 .
Ions
2+
Mg
Al3+
Si4+
P5+
S6+
Cl7+
Ar8+
K9+
Ca10+
Sc11+
Only 1P1
5
9
16
30
51
82
123
173
235
300
Total
105
135
168
204
244
287
334
384
438
496
Ions
M1
12+
5.58[−2]
2.14[−1]
7.64[−1]
2.52
7.68
21.6
56.1
136
308
661
Ti
V13+
Cr14+
Mn15+
Fe16+
Co17+
Ni18+
Cu19+
Zn20+
013416-6
Only 1P1
Total
M1
369
443
521
604
691
783
879
983
1091
558
624
695
774
850
936
1026
1125
1228
1.35[3]
2.64[3]
4.96[3]
9.00[3]
1.59[4]
2.73[4]
4.57[4]
7.49[4]
1.20[5]
THEORETICAL INVESTIGATION OF MAGNETIC-FIELD- . . .
PHYSICAL REVIEW A 88, 013416 (2013)
TABLE VI. 2p 5 3s 3P2 − 2p 6 1S0 M2 transition rates AM2 (in s−1 ) and magnetic-field-induced 2p 5 3s 3P2 − 2p 6 1S0 E1 transition rates
AMIT (M) (in s−1 ) for each of the magnetic sublevels M in the 2p5 3s 3P2 state under circumstances of B = 0.5, 1.5, and 2.5 T in Ne-like ions.
The average lifetimes τ (in s) for the 2p5 3s 3P2 state in the magnetic fields are given. Numbers in square brackets are the powers of 10.
B =0T
Ions
24
2+
Mg
Si4+
32 6+
S
40
Ar8+
40
Ca10+
48 12+
Ti
52
Cr14+
56
Fe16+
58
Ni18+
64
Zn20+
28
B = 0.5 T
B = 1.5 T
B = 2.5 T
AM2
AMIT (0)
AMIT (±1)
τ
AMIT (0)
AMIT (±1)
τ
AMIT (0)
AMIT (±1)
τ
7.63
1.12[2]
7.64[2]
3.45[3]
1.20[4]
3.52[4]
9.01[4]
2.08[5]
4.41[5]
8.82[5]
3.722
5.434
7.645
1.027[1]
1.324[1]
1.663[1]
2.035[1]
2.440[1]
2.861[1]
3.341[1]
2.791
4.076
5.733
7.700
9.931
1.247[1]
1.527[1]
1.830[1]
2.146[1]
2.506[1]
1.054[−1]
8.695[−3]
1.303[−3]
2.892[−4]
8.306[−5]
2.838[−5]
1.110[−5]
4.806[−6]
2.268[−6]
1.134[−6]
3.350[1]
4.891[1]
6.880[1]
9.240[1]
1.192[2]
1.497[2]
1.832[2]
2.196[2]
2.575[2]
3.007[2]
2.512[1]
3.668[1]
5.160[1]
6.930[1]
8.938[1]
1.122[2]
1.374[2]
1.647[2]
1.931[2]
2.255[2]
4.102[−2]
7.313[−3]
1.253[−3]
2.858[−4]
8.269[−5]
2.833[−5]
1.109[−5]
4.803[−6]
2.268[−6]
1.133[−6]
9.304[1]
1.359[2]
1.911[2]
2.567[2]
3.310[2]
4.157[2]
5.088[2]
6.100[2]
7.153[2]
8.352[2]
6.978[1]
1.019[2]
1.433[2]
1.925[2]
2.483[2]
3.118[2]
3.816[2]
4.575[2]
5.365[2]
6.264[2]
1.847[−2]
5.549[−3]
1.164[−3]
2.792[−4]
8.198[−5]
2.822[−5]
1.107[−5]
4.799[−6]
2.266[−6]
1.133[−6]
Using Eq. (23) and Eq. (24), we compute the MIT rates of
individual sublevels in the 2p5 3s 3P2 state under circumstances
of B = 0.5, 1.5, and 2.5 T, for most abundant isotopes without
nuclear spin. The results are presented in Table VI as well
as the 2p5 3s 3P2 − 2p6 1S0 M2 transition rates with B = 0 T.
In addition, we also display statistical average values τ of
lifetimes of the 2p5 3s 3P2 state in the external magnetic field,
which are obtained by
τ=
2J + 1
,
[AM2 + AMIT (M)]
(25)
where the summation is made over MIT and M2 decay
channels from all magnetic sublevels in the 2p5 3s 3P2 state.
As can be seen from Table VI, the external magnetic field
little influences the lifetime of the 2p5 3s 3P2 due to the
destructive contributions from 3P1 and 1P1 perturbers and
relatively large M2 transition rates, but still generates an
observable effect for ions near the neutral end. For instance,
the external magnetic field reduces the lifetime by 90% in a
B = 2.5 T magnetic field for Mg III.
strong electron correlations in neutral neon, we adopt a more
complicated computational model for taking into account
the correlation effects [9]. In Table VII the lifetimes of the
2p 5 3s 3P2 state for 20 Ne are presented in cases of B = 0.01
and 1 T, respectively. Compared with the 2p5 3s 3P2 − 2p6 1S0
M2 transition rate AM2 in the case of B = 0 T, we find from
this table that the external magnetic field, even fairly weak,
drastically reduces the lifetime of the 2p5 3s 3P2 state. In other
words, the lifetime of the 2p5 3s 3P2 state is highly sensitive
to the strength of the magnetic field. Hence the magnetic field
effect on the 2p5 3s 3P2 level lifetime could be part of the reason
behind the discrepancy.
In addition, we predict the magnetic-field-induced
2p5 3s 3P0 − 2p6 1S0 E1 transition rate for neutral neon. This
value reaches 95.2 s−1 in an 1 T external magnetic field, which
is much larger than the M1 rate, AM1 = 2.358 × 10−3 s−1 .
Moreover, the MIT rate is still comparable to the M1 even for
a field of only 0.005 T.
IV. SUMMARY
C. MITs in the case of Ne I
The accurate determination of lifetimes for metastable
states in the first excited configuration of rare gases is always
appealing [21]. For neutral neon, there still exists an unresolved
discrepancy in the lifetime of the 2p5 3s 3P2 state between
experiment and theory. The theoretical value [9] τ = 17.63 s
differs from the measurement [22] 14.73 s by 20%, which
is much larger than the uncertainties in both theory and
experiment. Therefore, we investigate the effect of an external
magnetic field on the lifetime of this level. Regarding the
To conclude, we have predicted rates for the magneticfield-induced 2p5 3s 3P0,2 − 2p6 1S0 E1 transitions using the
MCDHF method for Ne-like ions between Mg III and Zn XXI
without nuclear spin. We emphasize that both 2p5 3s 1P1 and
2p5 3s 3P1 perturber states are essential to include in order to
obtain reliable MIT rates. Using the reduced MIT rates ARMIT ,
reported in this paper, it is possible to predict rates for any
magnetic field strength. The atomic data presented in this paper
can be utilized for modeling plasma spectra. One should keep
in mind, however, there often exists the angular distribution
TABLE VII. 2p 5 3s 3P2 − 2p 6 1S0 M2 transition rates AM2 (in s−1 ) and magnetic-field-induced 2p5 3s 3P2 − 2p 6 1S0 E1 transition rates
AMIT (M) (in s−1 ) for each of the magnetic sublevels (M) in the 2p5 3s 3P2 state for 20 Ne without and with an external magnetic field of B = 0.01
T and 1 T, respectively. τ is an average level lifetime (in s) obtained with Eq. (25). Numbers in square brackets represent the powers of 10.
B =0T
Ions
20
Ne
B = 0.01 T
B =1T
AM2
AMIT (0)
AMIT (±1)
τ
AMIT (0)
AMIT (±1)
τ
5.672[−2]
1.197[−3]
8.979[−4]
17.45
11.97
8.979
1.655[−1]
013416-7
JIGUANG LI et al.
PHYSICAL REVIEW A 88, 013416 (2013)
of intensity for emission lines from plasma especially in the
presence of an external magnetic field. The relevant works are
ongoing.
We investigate the competition of the MITs with other
possible one-photon decay channels. It is found that the
magnetic-field-induced 2p5 3s 3P0 − 2p6 1S0 E1 transition is
the dominant decay channel for low-Z ions compared to the
2p5 3s 3P0 − 2p5 3s 3P1 M1 transition, while the influence of
magnetic fields on the lifetime of the 2p5 3s 3P2 state is small
but still observable for the ions at the neutral end of the
sequence.
In order to help resolve the discrepancy in the lifetime
of the 2p5 3s 3P2 state for neutral neon between experiment
and theory, the MIT rates in 20 Ne are calculated as well.
We find that the lifetime of both 2p5 3s 3P2 and 2p5 3s 3P0
states are extremely sensitive to the strength of magnetic fields.
Dependent on the experiment setup the effect of any magnetic
field could thus be a possible reason for this unresolved
discrepancy.
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ACKNOWLEDGMENTS
This work was supported by the National Natural Science
Foundation of China Grant No. 11074049, and by Shanghai
Leading Academic Discipline Project No. B107. We gratefully
acknowledge support from the Swedish Research Council
(Vetenskapsrådet) and the Swedish Institute under the Visby
program.
013416-8